Abstract

The idea of -convergence of real sequences was introduced by Kostyrko et al., (2000/01) and also independently by Nuray and Ruckle (2000). In this paper, we introduce the concepts of -statistical convergence of order and strong -Cesàro summability of order of real sequences and investigated their relationship.

1. Introduction

The idea of statistical convergence was given by Zygmund [1] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Steinhaus [2] and Fast [3] and later reintroduced by Schoenberg [4] independently. Later on it was further investigated from the sequence space point of view and linked with summability theory by Connor [5], Çınar et al. [6], Et et al. [7], Fridy [8], Güngör et al. [9, 10], Işik [11], Mohiuddine et al. [12–14], Mursaleen [15], Šalát [16], and many others.

The idea of -convergence of real sequences was introduced by Kostyrko et al. [17] and also independently by Nuray and Ruckle [18] (who called it generalized statistical convergence) as a generalization of statistical convergence. Later -convergence was studied by Das et al. [19–21], Kostyrko et al. [22], Mohiuddine et al. [23–25], Šalát et al. [26, 27], Tripathy and Hazarika [28, 29], and many others.

In this paper, we introduce the concepts of -statistical convergence of order and strong -Cesàro summability of order of real sequences and investigated their relationship. In Section 2 we give a brief overview about statistical convergence, strong -Cesàro summability, -convergence, and difference sequences. In Theorem 8, we give the inclusion relations between the sets of -convergent sequences and -summable sequences. In Theorem 10, we give the relationship between -summable sequences for different ’s. In Theorem 12, we give the relationship between the sets of -convergent sequences for different ’s.

2. Definition and Preliminaries

Let be the set of all sequences of real or complex numbers and let , and be, respectively, the Banach spaces of bounded, convergent, and null sequences with the usual norm , where , the set of positive integers. Also by , , , and , we denote the spaces of all bounded, convergent, and absolutely and -absolutely convergent series, respectively.

The definitions of statistical convergence and strong -Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set . The density of a subset of is defined by

where is the characteristic function of . It is clear that any finite subset of has zero natural density and .

The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in [30] and then statistical convergence of order and strong -Cesàro summability of order were studied by Çolak [31].

The notion of difference sequence spaces was introduced by Kızmaz [32] and it was generalized by Et et al. [33–35] such as
for , or , where , , , and so The sequence spaces are Banach spaces normed by
for , or . Let be any sequence spaces, if , then there exists one and only one such that
for sufficiently large ; for instance, . We use this fact in the following examples.

Recently, the difference sequence spaces have been studied in [10, 34, 36–39].

Let be nonempty set. Then a family of sets (power sets of ) is said to be an ideal if is additive, that is, implies , and hereditary; that is, , implies .

A nonempty family of sets is said to be a filter of if and only if (i) , (ii) implies , and (iii) , implies .

An ideal is called nontrivial if .

A nontrivial ideal is said to be admissible if .

If is a nontrivial ideal in , then the family of sets is a filter of , called the filter associated with .

Throughout the paper will stand for a nontrivial admissible ideal of .

We now introduce our main definitions.

Definition 1 (see [40]). A sequence is said to be -convergent if there exists such that, for all , the set . In this case, one writes . The set of all -convergent sequences will be denoted by .

Definition 2. Let be any real number. The sequence is said to be -statistical convergence of order (or -convergence) if there is a real number such that
In this case, we write or . The set of all -statistically convergent sequences of order will be denoted by . In the special case , we will write instead of .-statistical convergence of order is well defined for , but it is not well defined for in general. For this is defined as follows:
For every and we have
and so for any
we have
Therefore, the sequence is -statistically convergent of order , both to and ; that is, and . But this is impossible.

It is easy to see that every -convergent sequence is -statistically convergent of order , but converse does not hold. For this, consider a sequence defined by

It is clear that for , but .

Definition 3. Let be any real number and let be a positive real number. A sequence is said to be strong -Cesàro summable of order (or strong -summable) if there is a real number such that
In this case, we write . The set of all strong -Cesàro summable sequences of order to will be denoted by . In the special case , we will write instead of .

3. Main Results

In this section, we give the main results of this paper. In Theorem 8 we give the inclusion relations between the sets of -convergent sequences and -summable sequences. In Theorem 10, we give the relationship between -summable sequences for different ’s. In Theorem 12, we give the relationship between the sets of - convergent sequences for different ’s.

Theorem 4. Let be any real number and suppose that , , and ; then(i),
(ii).

Proof. (i) Suppose that and ; then
and so .(ii) Now suppose that and ; then we have
and so . Now for all , we have
Then
Hence .The proofs of the following two theorems are easy and thus omitted.

Theorem 5. Let be any real number; then the limit of any -convergent sequence is uniquely determined.

Theorem 6. Let , , and be real sequences such that . If , then .

The proof of the following theorem is obtained by using the same techniques of Savas and Das [21, Theorem 2.4]; therefore we give it without proof.

Theorem 7. Let be any real number, then is a closed subset of .

In the following theorem we investigate the relationship between -statistically convergent sequences and strong -summable sequences.

Theorem 8. Let and be fixed real numbers such that , and let be a positive real number; then , and the inclusion is strict.

Proof. Let and ; then we can write
and so
Then for any , we have
This completes the proof.

Taking , we show the strictness of the inclusion for a special case. For this, consider the sequence defined by

For every and we have

and for any we get
Since the set on the right-hand side is a finite set and so belongs to , it follows that for . On the other hand, for we have
Then
for some which belongs to , since is admissible. So .

The converse of Theorem 8 does not hold, in general. To show this, we must find a sequence that is -bounded and -convergent, but need not to be -summable. For this, consider a sequence defined by (10). It can be shown that and for and for . Therefore, for .

Corollary 9. If a sequence is -convergent to , then it is -convergent to .

Theorem 10. Let , and let be a positive real number; then and the inclusion is strict.

Proof. The inclusion part of proof is trivial. Taking , we show the strictness of the inclusion for a special case. Define the sequence such that
It can easily be shown that
but
So for but for .The following result is a consequence of Theorem 10.

Corollary 11. Let be a positive real number. Then(i)if , then ,(ii) for each .

Theorem 12. Let and be fixed real numbers such that ; then , and the inclusion is strict.

Proof. Let . Then given and such that , we may write
and this gives that .We show the strictness of the inclusion for a special case. Define the sequence such that
Then for , but for .The following result is a consequence of Theorem 10.

Corollary 13. Let be a real number; then .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.